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We propose a quantum soft-covering problem for a given general quantumchannel and one of its output states, which consists in finding the minimumrank of an input state needed to approximate the given channel output. We thenprove a one-shot quantum covering lemma in terms of smooth min-entropies byleveraging decoupling techniques from quantum Shannon theory. This coveringresult is shown to be equivalent to a coding theorem for rate distortion undera posterior (reverse) channel distortion criterion [Atif, Sohail, Pradhan,arXiv:2302.00625]. Both one-shot results directly yield corollaries about thei.i.d. asymptotics, in terms of the coherent information of the channel.  The power of our quantum covering lemma is demonstrated by two additionalapplications: first, we formulate a quantum channel resolvability problem, andprovide one-shot as well as asymptotic upper and lower bounds. Secondly, weprovide new upper bounds on the unrestricted and simultaneous identificationcapacities of quantum channels, in particular separating for the first time thesimultaneous identification capacity from the unrestricted one, proving along-standing conjecture of the last author.

Quantum soft-covering lemma with applications to rate-distortion coding, resolvability and identification via quantum channels